The parity anomaly of a single massless Dirac fermion is manifested as a half-quantized Hall response once the required parity-breaking regularization is supplied. We study the corresponding response for two-dimensional chiral Lifshitz fermions. For a node with low-energy coupling ${(q_x+\mathrm{i}{q}_y)}^N$, the Fermi-loop winding number $W=N$ gives an anomalous Hall response ${\sigma }_{xy}^{\mathrm{anom}}=(W/2)e^2/h$. We examine this mechanism in chiral multilayer Haldane models tuned to a valley-selective critical line, where a single gapless chiral Lifshitz fermion remains at the $K$ valley. For the AB, ABC, and ABCA stacks, the projected pseudospin textures give $W=2,3,4$, while Kubo calculations yield ${\sigma }_{xy}=(1,3/2,2)e^2/h$; integrated equilibrium boundary currents reproduce the same $W/2$ sequence. Six-terminal simulations with dephasing further show that the Hall conductivity is quantized at $W/2$ while the longitudinal conductivity remains finite. These results identify gapless chiral Lifshitz fermions as a higher-order, winding-controlled generalization of the parity-anomalous Hall response, distinct from a quantum anomalous Hall insulating state even when $W/2$ is an integer.

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• Anomalous Hall effect
• Transport phenomena
• Nonequilibrium Green's function